Question: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle CBD = 9x + 33$, and $ m \angle ABC = 6x - 3$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {6x - 3} + {9x + 33} = {90}$ Combine like terms: $ 15x + 30 = 90$ Subtract $30$ from both sides: $ 15x = 60$ Divide both sides by $15$ to find $x$ $ x = 4$ Substitute $4$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 9({4}) + 33$ Simplify: $ {m\angle CBD = 36 + 33}$ So ${m\angle CBD = 69}$.